On Nov 25, 12:55 am, Graham Cooper <grahamcoop...@gmail.com> wrote:> On Nov 25, 6:37 pm, WM <mueck...@rz.fh-augsburg.de> wrote:>>>>>>>>>> > Matheology § 161>> > {{Yet another application of set theory?}} I propose here, then, first> > to illustrate, and then to discuss theoretically, the nature and ideal> > outcome of any recurrent operation of thought, and to develope, in> > this connection, what one may call the positive nature of the concept> > of Infinite Multitude.>> > Prominent among the later authors who have dealt with our problem from> > the mathematical side, is George Cantor. [...] With this theory of the> > Mächtigkeiten I shall have no space to deal in this paper, but it is> > of great importance for forming the conception of the determinate> > Infinite.>> > A map of England, contained within England, is to represent, down to> > the minutest detail, every contour and marking, natural or artificial,> > that occurs upon the surface of England.>> > Our map of England, contained in a portion of the surface of England,> > involves, however, a peculiar and infinite development of a special> > type of diversity within our map. For the map, in order to be> > complete, according to the rule given, will have to contain, as a part> > of itself, a representation of its own contour and contents. In order> > that this representation should be constructed, the representation> > itself will have to contain once more, as a part of itself, a> > representation of its own contour and contents; and this> > representation, in order to be exact, will have once more to contain> > an image of itself; and so on without limit. We should now, indeed,> > have to suppose the space occupied by our perfect map to be infinitely> > divisible, even if not a continuum.>> > That such an endless variety of maps within maps could not physically> > be constructed by men, and that ideally such a map, if viewed as a> > finished construction, would involve us in all the problems about the> > infinite divisibility of matter and of space, I freely recognize.>> > Suppose that, for an instance, we had accepted this assertion as true.> > Suppose that we then attempted to discover the meaning implied in this> > one assertion. We should at once observe that in this one assertion,> > "A part of England perfectly maps all England, on a smaller scale,"> > there would be implied the assertion, not now of a process of trying> > to draw maps, but of the contemporaneous presence, in England, of an> > infinite number of maps, of the type just described. The whole> > infinite series, possessing no last member, would be asserted as a> > fact of existence.>> > We should, moreover, see how and why the one and the infinitely many> > are here, at least within thought's realm, conceptually linked. Our> > map and England, taken as mere physical existences, would indeed> > belong to that realm of "bare external conjunctions." Yet the one> > thing not externally given, but internally self-evident, would be that> > the one plan or purpose in question, namely, the plan fulfilled by the> > perfect map of England, drawn within the limits of England, and upon a> > part of its surface, would, if really expressed, involve, in its> > necessary structure, the series of maps within maps such that no one> > of the maps was the last in the series.>> > This way of viewing the case suggests that, as a mere matter of> > definition, we are not obliged to deal solely with processes of> > construction as successive, in order to define endless series. A> > recurrent operation of thought can be characterized as one that, if> > once finally expressed, would involve, in the region where it had> > received expression, an infinite variety of serially arranged facts,> > corresponding to the purpose in question.>> > [Josiah Royce: "The world and the individual", MacMillan, London> > (1900) p. 500ff]http://www.archive.org/stream/worldindividual00royciala#page/n0/mode/......>> > The repeated application of the fotocopier has been proposed as a> > cheap replacement for expensive electron microscopes. Unfortunately I> > have forgotten the name of the inventor of this idea.>> > Regrads, WM>> there are no mathematicians reading.>> logic is a study of literature only, no different to knowing vast> journals of legal precedents, the entire art history of the Incubus> period, the works of Shakespear or illustrated Medical Journals.>> You are talking to Clayton's Arts Graduates who wish to further the> 'study' of logic, which opposes the _application_ of logic.>> --> if( if(t(S),f(R)) , if(t(R),f(S)) ).> if it's sunny then it's not raining> ergo> if it's raining then it's not sunny>> S: if stops(S) gosub S> G. GREENE: this proves stops() must be un-computable!> SCI.LOGIC